I don't have an iPad Mini, but I do have an iPad 2 and I know for a fact that it can at least play 720p HD movies.
However, all my movies are ripped and converted from my Blu-ray collection and so, I have the SD, 720p HD, and 1080p HD versions of each movie for all my devices to enjoy. iTunes automatically chooses the correction version to sync for my iDevice.
For example, if I hook up my iPad 3rd Gen, it'll sync the 1080p HD version. If I hook up my iPhone 5 or iPad 2, the 720p HD version is synced. Then, anything can sync the SD version if I so choose.